Dedicated to $e^{i\pi} + 1729$

Number Theory Level 5

$\left(\sum_{n=1}^{10000} 2S_m + 5S_n\right)^{1729}$

Let $S_n$ and $S_m$ be two natural numbers for arbitary natural numbers $n$ and $m$ : such that , $S_{n-1}=2S_{m+1}$ .Also $5^m | S_m$ , provided that $(n,m) \leq 10000$ .

Then evaluate above sumation modulo 10

Bonus : Generalize this.

Dedicated to eiπ+1729e^{i\pi} + 1729eiπ+1729

Dedicated to $e^{i\pi} + 1729$